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Introduction to Linear Algebra with Mathematica

Glossary

Laplace Transform

Historically, the Laplace transform was first introduced by Oliver Heaviside, a British mathematician and electrical engineer who is considered as a father of operational calculus. He also contributed to Electromagnetic Theory; in particular, he simplified the Maxwell's equations to the form that is used today.

Unfortunately, Heaviside by himself was unable to explain how he used the Laplace transformation for solving differential equations---this is a typical case with genius people. Therefore, mathematicians called Oliver's technique after Pierre-Simon Laplace based on a special integral that appeared in his work, but who did not explore the notion of transformation.

We will deal with complex-valued functions f(x) of real variable x ∈ ℝ such that f(x) ≡ 0 for x < 0. We also assume that the product of f(x) and exponential function e−αx, for some real α, belong to the Banach space 𝔏¹. The its Fourier transform is a bounded continuous function on whole space ℝ; however, the Fourier transform of function f(x) may not exist. On the other hand, the Fourier transform of function f exists for complex argument ξ:

\begin{align*} f^F (\xi ) &= ℱ\left[ f(x) \right]_{x\to \xi} = \mbox{PV} \int_{-\infty}^{\infty} f(x)\,e^{-{\bf j}\xi x} {\text d} x \\ &= \int_0^{\infty} f(x)\,e^{-{\bf j}\xi x} {\text d} x , \qquad \Im\xi \le -\alpha . \end{align*} Here VP stands for Cauchy principal value and j (also denoted by ⅉ) is the imaginary unit on complex plane ℂ, so j² = −1. We have
\[ \left\vert f(x)\,e^{-{\bf j}\xi x} \right\vert = \left\vert f(x) \right\vert e^{\Im \xi x} \leqslant \left\vert f(x) \right\vert e^{-\alpha x} . \]
Moreover, the Fourier transform fF(ξ) = ℱ[f](ξ) is an analytic function in half plane Imξ ≤ −α. Therefore, its inverse can be evaluated over any horizontal line (parallel to the real axis) within this half plane:
\[ f(x) = ℱ^{-1}\left[ f^F (\xi ) \right]_{\xi\to x} = \mbox{PV} \int_{\gamma{\bf j}--\infty}^{\gamma{\bf j}+\infty} f^F (\xi ) \,e^{{\bf j} \xi x} {\text d}\xi . \]
Let us consider complex-valued functions φ(x), defined on real axis that satisfy the condition
\[ \left\vert \varphi \right\vert \leqslant C\, e^{-\alpha x} \quad \mbox{for}\ x>0 \quad \mbox{and} \quad \varphi(x) \equiv 0 \quad x < 0 . \]
Such functions are called functions-originals in the theory of Laplace transformation. Fourier transformation of function-original
\[ f^F (\xi ) = ℱ\left[ f \right]_{x\to \xi} = \int_{0}^{\infty} f(x)\,e^{-{\bf j}\xi x} {\text d} x \]
exists in the half-plane Imξ < −α. If we set λ = ⅉξ, we arrive at
\begin{equation} \label{EqLaplace.1} f^L (\xi ) = ℒ\left[ f \right]_{x\to \lambda} = \int_{0}^{\infty} f(x)\,e^{-\lambda x} {\text d} x . \end{equation}
This function ℒ[f](λ) = fL(λ) is called the Laplace transform of function f. The Laplace transform fL(λ) is an analytic function in the domain Reλ > α. The following condition is known as the Riemann-Lebesgue lemma:
\[ \lim_{|\Im\lambda |\to\infty} f^L (\lambda ) = 0 . \]
This limit is uniform in the strip α < 𝑎 ≤ Reλ ≤ b. The inverse Laplace transform is defined by the Bromwich integral:
\begin{equation} \label{EqLaplace.2} f(t) = ℒ^{-1} \left[ f^L (\lambda ) \right]_{\lambda \to t} = \lim_{T\to\infty} \frac{1}{2\pi {\bf j}} \int_{\gamma - {\bf j}T}^{\gamma + {\bf j}T} e^{\lambda t} f^L (\lambda )\,{\text d}\lambda . \end{equation}
   ■
End of Example 1
ℒ ℒ ℒ ℒ ℒ

 

  1. Asmar, N.H., Partial Differential Equations with Fourier Series and Boundary Value Problems: Third Edition (Dover Books on Mathematics), 3026.
  2. Debnath, L. and Bhatta, D., Integral Transformes and theor Applications, Second Edition, Chapman and Hall/CRC, 2006.
  3. Davis, B., Integral transforms and their Applications, Third edition, Springer.
  4. Evans, L.C., Partial Differential Equations (Graduate Studies in Mathematics), Amer Mathematical Society; 2nd edition, 2010.
  5. Haberman, R., Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Pearson; 5th edition, 2012.
  6. Pikulin, V.P. and Pohozaev, S.I., Equations in Mathematical Physics: A practical course, Birkhäuser; 2001st edition, Springer, 2012.

 

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